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## G = (C2×C4).D12order 192 = 26·3

### 3rd non-split extension by C2×C4 of D12 acting via D12/C3=D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C2×C4).D12
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C6×Q8 — C12.23D4 — (C2×C4).D12
 Lower central C3 — C6 — C2×C6 — C2×C12 — (C2×C4).D12
 Upper central C1 — C2 — C22 — C2×Q8 — C4.10D4

Generators and relations for (C2×C4).D12
G = < a,b,c,d | a2=b4=1, c12=b2, d2=b, cbc-1=ab=ba, cac-1=dad-1=ab2, bd=db, dcd-1=b-1c11 >

Subgroups: 240 in 64 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, D6, C2×C6, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, C3⋊C8, C24, D12, C2×Dic3, C2×C12, C3×Q8, C22×S3, C4.10D4, C4.10D4, C4.4D4, C4.Dic3, C4×Dic3, D6⋊C4, C3×M4(2), C2×D12, C6×Q8, C42.C4, C12.10D4, C3×C4.10D4, C12.23D4, (C2×C4).D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C23⋊C4, D6⋊C4, C42.C4, C23.6D6, (C2×C4).D12

Character table of (C2×C4).D12

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D size 1 1 2 24 2 4 4 4 12 12 2 4 8 8 24 24 4 4 8 8 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 1 1 -1 1 1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ5 1 1 1 1 1 -1 -1 1 -1 -1 1 1 i -i -i i -1 -1 -1 1 -i -i i i linear of order 4 ρ6 1 1 1 -1 1 -1 -1 1 1 1 1 1 i -i i -i -1 -1 -1 1 -i -i i i linear of order 4 ρ7 1 1 1 1 1 -1 -1 1 -1 -1 1 1 -i i i -i -1 -1 -1 1 i i -i -i linear of order 4 ρ8 1 1 1 -1 1 -1 -1 1 1 1 1 1 -i i -i i -1 -1 -1 1 i i -i -i linear of order 4 ρ9 2 2 2 0 2 -2 2 -2 0 0 2 2 0 0 0 0 -2 -2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 0 -1 2 2 2 0 0 -1 -1 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 2 2 0 -1 2 2 2 0 0 -1 -1 -2 -2 0 0 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ12 2 2 2 0 2 2 -2 -2 0 0 2 2 0 0 0 0 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 0 -1 -2 2 -2 0 0 -1 -1 0 0 0 0 1 1 -1 1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ14 2 2 2 0 -1 -2 2 -2 0 0 -1 -1 0 0 0 0 1 1 -1 1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ15 2 2 2 0 -1 -2 -2 2 0 0 -1 -1 -2i 2i 0 0 1 1 1 -1 -i -i i i complex lifted from C4×S3 ρ16 2 2 2 0 -1 -2 -2 2 0 0 -1 -1 2i -2i 0 0 1 1 1 -1 i i -i -i complex lifted from C4×S3 ρ17 2 2 2 0 -1 2 -2 -2 0 0 -1 -1 0 0 0 0 -1 -1 1 1 -√-3 √-3 √-3 -√-3 complex lifted from C3⋊D4 ρ18 2 2 2 0 -1 2 -2 -2 0 0 -1 -1 0 0 0 0 -1 -1 1 1 √-3 -√-3 -√-3 √-3 complex lifted from C3⋊D4 ρ19 4 4 -4 0 4 0 0 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from C23⋊C4 ρ20 4 4 -4 0 -2 0 0 0 0 0 -2 2 0 0 0 0 2√-3 -2√-3 0 0 0 0 0 0 complex lifted from C23.6D6 ρ21 4 4 -4 0 -2 0 0 0 0 0 -2 2 0 0 0 0 -2√-3 2√-3 0 0 0 0 0 0 complex lifted from C23.6D6 ρ22 4 -4 0 0 4 0 0 0 -2i 2i -4 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42.C4 ρ23 4 -4 0 0 4 0 0 0 2i -2i -4 0 0 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C42.C4 ρ24 8 -8 0 0 -4 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal faithful, Schur index 2

Smallest permutation representation of (C2×C4).D12
On 48 points
Generators in S48
(2 14)(4 16)(6 18)(8 20)(10 22)(12 24)(26 38)(28 40)(30 42)(32 44)(34 46)(36 48)
(1 45 13 33)(2 46 14 34)(3 35 15 47)(4 36 16 48)(5 25 17 37)(6 26 18 38)(7 39 19 27)(8 40 20 28)(9 29 21 41)(10 30 22 42)(11 43 23 31)(12 44 24 32)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48)
(1 32 45 12 13 44 33 24)(2 11 46 43 14 23 34 31)(3 30 35 22 15 42 47 10)(4 21 36 41 16 9 48 29)(5 28 25 8 17 40 37 20)(6 7 26 39 18 19 38 27)

G:=sub<Sym(48)| (2,14)(4,16)(6,18)(8,20)(10,22)(12,24)(26,38)(28,40)(30,42)(32,44)(34,46)(36,48), (1,45,13,33)(2,46,14,34)(3,35,15,47)(4,36,16,48)(5,25,17,37)(6,26,18,38)(7,39,19,27)(8,40,20,28)(9,29,21,41)(10,30,22,42)(11,43,23,31)(12,44,24,32), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,32,45,12,13,44,33,24)(2,11,46,43,14,23,34,31)(3,30,35,22,15,42,47,10)(4,21,36,41,16,9,48,29)(5,28,25,8,17,40,37,20)(6,7,26,39,18,19,38,27)>;

G:=Group( (2,14)(4,16)(6,18)(8,20)(10,22)(12,24)(26,38)(28,40)(30,42)(32,44)(34,46)(36,48), (1,45,13,33)(2,46,14,34)(3,35,15,47)(4,36,16,48)(5,25,17,37)(6,26,18,38)(7,39,19,27)(8,40,20,28)(9,29,21,41)(10,30,22,42)(11,43,23,31)(12,44,24,32), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48), (1,32,45,12,13,44,33,24)(2,11,46,43,14,23,34,31)(3,30,35,22,15,42,47,10)(4,21,36,41,16,9,48,29)(5,28,25,8,17,40,37,20)(6,7,26,39,18,19,38,27) );

G=PermutationGroup([[(2,14),(4,16),(6,18),(8,20),(10,22),(12,24),(26,38),(28,40),(30,42),(32,44),(34,46),(36,48)], [(1,45,13,33),(2,46,14,34),(3,35,15,47),(4,36,16,48),(5,25,17,37),(6,26,18,38),(7,39,19,27),(8,40,20,28),(9,29,21,41),(10,30,22,42),(11,43,23,31),(12,44,24,32)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)], [(1,32,45,12,13,44,33,24),(2,11,46,43,14,23,34,31),(3,30,35,22,15,42,47,10),(4,21,36,41,16,9,48,29),(5,28,25,8,17,40,37,20),(6,7,26,39,18,19,38,27)]])

Matrix representation of (C2×C4).D12 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 72 0 0 0 1 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 46 54 0 0 0 0 0 27 0 0 0 0 46 27 0 27 0 0 46 27 27 0
,
 0 1 0 0 0 0 72 1 0 0 0 0 0 0 1 0 71 0 0 0 0 0 72 1 0 0 60 0 72 0 0 0 14 46 72 0
,
 72 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 71 0 0 0 1 0 72 72 0 0 14 46 72 0 0 0 60 0 72 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,46,46,0,0,54,27,27,27,0,0,0,0,0,27,0,0,0,0,27,0],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,0,60,14,0,0,0,0,0,46,0,0,71,72,72,72,0,0,0,1,0,0],[72,0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,14,60,0,0,0,0,46,0,0,0,71,72,72,72,0,0,0,72,0,0] >;

(C2×C4).D12 in GAP, Magma, Sage, TeX

(C_2\times C_4).D_{12}
% in TeX

G:=Group("(C2xC4).D12");
// GroupNames label

G:=SmallGroup(192,36);
// by ID

G=gap.SmallGroup(192,36);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,422,184,1123,794,297,136,851,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=1,c^12=b^2,d^2=b,c*b*c^-1=a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*d=d*b,d*c*d^-1=b^-1*c^11>;
// generators/relations

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